Modern Control Engineering

562 Chapter 7 / Control Systems Analysis and Design by the Frequency-Response Method

B–7–2.Consider the system whose closed-loop transfer

function is

Obtain the steady-state output of the system when it is sub-

jected to the input r(t)=Rsinvt.

B–7–3.Using MATLAB, plot Bode diagrams of G 1 (s)and

G 2 (s)given below.

G 1 (s)is a minimum-phase system and G 2 (s)is a nonmini-

mum-phase system.

B–7–4.Plot the Bode diagram of

B–7–5.Given

show that

B–7–6.Consider a unity-feedback control system with the

following open-loop transfer function:

This is a nonminimum-phase system. Two of the three

open-loop poles are located in the right-half splane as

follows:

Plot the Bode diagram of G(s)with MATLAB. Explain why

the phase-angle curve starts from 0° and approaches ±180°.

s =0.2328-j0.7926

s =0.2328+j0.7926

Open-loop poles at s=-1.4656

G(s)=

s+0.5

s^3 +s^2 + 1

@GAjvnB@=

1

2 z

G(s)=

v^2 n

s^2 + 2 zvn s+v^2 n

G(s)=

10 As^2 +0.4s+ 1 B

sAs^2 +0.8s+ 9 B

G 2 (s)=

1 - s

1 +2s

G 1 (s)=

1 +s

1 +2s

C(s)

R(s)

=

KAT 2 s+ 1 B

T 1 s+ 1

B–7–7.Sketch the polar plots of the open-loop transfer

function

for the following two cases:

(a)

(b)

B–7–8.Draw a Nyquist locus for the unity-feedback control

system with the open-loop transfer function

Using the Nyquist stability criterion, determine the stabili-

ty of the closed-loop system.

B–7–9.A system with the open-loop transfer function

is inherently unstable. This system can be stabilized by adding

derivative control. Sketch the polar plots for the open-loop

transfer function with and without derivative control.

B–7–10.Consider the closed-loop system with the following

open-loop transfer function:

Plot both the direct and inverse polar plots of G(s)H(s)

withK=1andK=10. Apply the Nyquist stability crite-

rion to the plots, and determine the stability of the system

with these values of K.

B–7–11.Consider the closed-loop system whose open-loop

transfer function is

Find the maximum value of Kfor which the system is stable.

B–7–12.Draw a Nyquist plot of the following G(s):

B–7–13.Consider a unity-feedback control system with the

following open-loop transfer function:

Draw a Nyquist plot of G(s)and examine the stability of

the system.

G(s)=

1

s^3 +0.2s^2 +s+ 1

G(s)=

1

sAs^2 +0.8s+ 1 B

G(s)H(s)=

Ke-2s

s

G(s)H(s)=

10K(s+0.5)

s^2 (s+2)(s+10)

G(s)H(s)=

K

s^2 AT 1 s+ 1 B

G(s)=

K(1-s)

s+ 1

T 7 Ta 7 0, T 7 Tb 70

Ta 7 T 7 0, Tb 7 T 70

G(s)H(s)=

KATa s+ 1 BATb s+ 1 B

s^2 (Ts+1)

Openmirrors.com